MAXWELL'S EQUATIONS FROM ELECTROSTATICS AND EINSTEIN'S

GRAVITATIONAL FIELD EQUATION FROM NEWTON'S

UNIVERSAL LAW OF GRAVITATION

USING TENSORS

Michael E. Burns, B.S.

Problem in Lieu of Thesis Prepared for the Degree of

MASTER OF SCIENCE

UNIVERSITY OF NORTH TEXAS

May 2004

APPROVED: Donald H. Kobe, Major Professor Jacek M. Kowalski, Committee Member Carlos A. Ordonez, Committee Member Duncan Weathers, Program Coordinator Floyd M. McDaniel, Chair of the

Department of Physics Sandra L. Terrell, Interim Dean of the

Robert B. Toulouse School of Graduate Studies

Burns, Michael E., Maxwell's Equations from Electrostatics and Einstein's

Gravitational Field Equation from Newton's Universal Law of Gravitation Using Tensors.

Master of Science (Physics), May 2004, 87 pp., references, 27 titles.

Maxwell's equations are obtained from Coulomb's Law using special relativity.

For the derivation, tensor analysis is used, charge is assumed to be a conserved scalar,

the Lorentz force is assumed to be a pure force, and the principle of superposition is

assumed to hold.

Einstein's gravitational field equation is obtained from Newton's universal law of

gravitation. In order to proceed, the principle of least action for gravity is shown to be

equivalent to the maximization of proper time along a geodesic. The conservation of

energy and momentum is assumed, which, through the use of the Bianchi identity,

results in Einstein's field equation.

ii

TABLE OF CONTENTS

Page Chapter

1. INTRODUCTION ................................................................................................... 1

2. SPECIAL RELATIVITY: DERIVATION OF MAXWELL'S EQUATIONS.............................................................................................. 3

2.1. Introduction ......................................................................................... 3 2.2. Lorentz Transformations ..................................................................... 4 2.3. Tensors............................................................................................. 10

2.3a. Scalar Fields ........................................................................ 11 2.3b. Vector Fields ........................................................................ 11 2.3c. Second Rank Tensors.......................................................... 15 2.3d. Superscripts vs. Subscripts .................................................. 19

2.4. Generalizing Electrostatics to the Time Dependent Gauss' Law and the Ampere Maxwell Law.................................... 21

2.5. Generalizing Electrostatics to Faraday's Law and the Law of No Magnetic Monopoles .............................................. 27

2.6. The Lorentz Force............................................................................. 30 2.7. Conclusion ........................................................................................ 33

3. GENERAL RELATIVITY: EINSTEIN'S GRAVITATIONAL FIELD EQUATION FROM NEWTON'S LAW OF UNIVERSAL GRAVIATION ..................................................................... 35

3.1. Introduction ....................................................................................... 35 3.2. Poisson's Equation for Gravity .......................................................... 38 3.3. Space-time Curvature ....................................................................... 40 3.4. Principle of Least Action ................................................................... 44 3.5. Einstein's Equation ........................................................................... 46 3.6. Conclusion ........................................................................................ 49

4. CONCLUSION..................................................................................................... 50

APPENDICES ............................................................................................................... 51 REFERENCES.............................................................................................................. 84

1

CHAPTER 1

INTRODUCTION

Thousands of years ago, the more advanced ancient societies were aware of the

forces of electricity and magnetism.1 Several hundreds of years ago, these forces were

considered to be different manifestations of the same kind of force. Ironically, this

unification was dismissed for centuries as the scientific community pursued a more

enlightened philosophy. It wasn't until the time of James Clerk Maxwell that the

unification of these two forces was again accepted. Of course, this did not play well

with the mechanics proposed by Isaac Newton.2 The two theories, electromagnetism

and classical mechanics, required different treatments when transformed from one

coordinate system to another. This indicated that at least one of the two theories

needed an adjustment. The first idea was to adjust Maxwell's equations to

accommodate classical mechanics, but Albert Einstein saw things the other way

around.2

Einstein used symmetry to show algebraically that length contraction and time

dilation are consequences of the invariance of the speed of light.3 To solidify the

principle of relativity, he developed a geometrical picture in which tensors that transform

according to Lorentz transformation matrices represent all physical objects in flat space-

time.2 In this regard, the magnetic field is a consequence of the application of special

relativity to the electric field under a Lorentz transformation. Ultimately, tensor analysis

can be used to derive Maxwell's equations from Coulomb's Law and the assumptions

that charge is a conserved scalar and that the principle of superposition holds.2

2

After completing his theory of special relativity, Einstein realized that the next

great accomplishment would be to make gravitation consistent with the special theory of

relativity.2 Mass, the source of gravitation, is very different than charge, and the

analogy between gravity and electrostatics breaks down.2 Einstein showed that the

gravitational field is actually a distortion of space-time itself.2 For general relativity,

tensors had to be used so that the form of the equations of physical laws remained

invariant under general coordinate transformations. This is called the principle of

general covariance.4

Applying the principle of general covariance and the equivalence principle to

Newton's universal law of gravitation gives Einstein's gravitational field equation, if

energy is conserved. Poisson's equation for gravity can be generalized to Einstein's

gravitational field equation, a second rank tensor equation.2 In the limit that the sources

are stationary and the fields they produce are weak, Einstein's equation reduces to

Poisson's equation for gravity.2 Therefore, Newton's law for gravity, which he

presented5 in 1686, is still a valid way to describe "weak gravitational interactions" in

which the interacting bodies have constant mass and have negligible velocity compared

to the speed of light. That Newton's law for gravity is still valid today in the appropriate

limit demonstrates the smooth progression of classical physics over the last few

centuries.

3

CHAPTER 2

SPECIAL RELATIVITY: DERIVATION OF MAXWELL'S EQUATIONS

2.1. Introduction

Magnetism was most likely first recognized thousands of years ago by the

Chinese as a force, though most thought it to be of magical origin.1 The source of this

magic was known as lodestone. Almost as long ago, the Greeks first recognized that a

force emanated from amber. This was the first observation of the electrostatic force.1 A

millennium later, in the sixteenth century during the renaissance, interest in science

grew, and sparked an explosion of scientific discoveries. Charles Coulomb

experimentally demonstrated in 1785 that the electrostatic force obeys an inverse

square law.1 In 1873, James Clerk Maxwell presented his Treatise on Electricity and

Magnetism which showed a unification of the contemporary theories of electricity and

magnetism into an elegant set of four equations known as Maxwell's equations.6 Along

with the elegance of the equations came the realization that electromagnetic waves

should propagate away from a source at the speed of light (a phenomenon that Heinrich

Hertz verified experimentally in a series of experiments conducted between 1879 and

1889).6 In 1890, Hendrik A. Lorentz proposed transformations that maintained the form

of Maxwell's equations for frames of reference moving at different velocities with respect

to the source of the electromagnetic field.7 A transformation of this type is known as a

Lorentz transformation. Einstein showed in 1905 that the Lorentz transformations could

be derived algebraically with the assumption that the speed of light is the same for any

observer in any inertial frame of reference,3 and so modified classical mechanics to be

consistent with electromagnetism.2

4

There have been many attempts to obtain Maxwell's equations from Coulomb's

Law using special relativity, though the approaches and assumptions vary. R. P.

Feynman uses the scalar and vector potentials.8 E. Krefetz indicates that a multitude of

assumptions must be made in order to carry out the derivation.9 W. Rindler gives a very

similar approach, as well as using potentials to justify certain steps.10 D. H. Frisch and

L. Wilets offer a very detailed and rigorous approach.11 One of the most significant

approachs to the one given in this chapter was presented by D. H. Kobe.2 However,

there are some considerations that are given more attention in this chapter.

Tensors are discussed in general in Sec. 3. In Sec. 4 Gauss' Law for

electrostatics is generalized to Gauss' Law for time varying fields and to the Ampere-

Maxwell Law. In Sec. 5 the conservative nature of the electrostatic field is generalized

to Faraday's Law and the law of no magnetic monopoles. The proper time introduced in

Sec. 3 and the Faraday tensor developed in Secs. 4 and 5 are used to generalize

Newton's second law for a charged particle to the Lorentz force in Sec. 6. Sec. 7 gives

the conclusion for this chapter.

2.2. Lorentz Transformations

Einstein postulated that the speed of light should be the same for both of two

observers, regardless of their relative motion (as long as neither experiences an

acceleration). Using this postulate, the Lorentz transformations can be obtained.

Consider two observers, A and B, moving along the x -axis. They both pass through

the origin at time 0=t . Let A observe B to move to the right (in the x+ direction) at a

constant speed, v . Now, assume that A sends out a light pulse isotropically from the

origin at 0=t . Of course, observer A will consider this as a spherical wavefront,

5

centered at herself, the radius of which increases at a rate, c . The "strange"

consequence of Einstein's postulate is that B will also consider this as a spherical

wavefront, centered at himself, the radius of which also increases at a rate, c . This is

strange in terms of Galilean relativity. While B should consider a spherical wavefront,

he should consider the center of the sphere to move to the left (in the x− direction).

However, according to Einstein's special theory of relativity, the center of the sphere will

not move at all with respect to B; B will also remain at the center of the sphere.

Therefore, some aspect of the common sense of Galilean relativity must change.

Einstein's relativity abandons the notions of absolute time and absolute space, and it

replaces them with the invariance of the speed of light in all inertial frames. The Lorentz

transformations, instead of Galilean transformations, determine Cartesian coordinate

transformations from one inertial frame to another.

To proceed with the derivation, it is convenient to introduce two coordinate

systems, K and K', and restrict the discussion to 1 spatial dimension. Let K be defined

as the coordinate system in which A is at rest and B is moving to the right at a constant

velocity, v . Let K' be defined as the coordinate system in which B is at rest. Since B is

moving to the right relative to A at a speed, v , then A moves to the left relative to B at a

speed, v . Therefore, in K', A is moving to the left at a constant velocity, v . This has a

straightforward representation in Cartesian coordinates.

In coordinate system K, let Ax be the value of the x coordinate of A's position,

and let Bx be the value of the x coordinate of B's position. Thena

0=Ax , (1) a Note: the coordinates are, in general, functions of time.

6

and

vtxB = . (2)

In coordinate system K', let Ax′ be the value of the x coordinate of A's position,

and let Bx′ be the value of the x coordinate of B's position.b Then

tvxA ′−=′ , (3)

and

0=′Bx . (4)

Since the speed of light is invariant, the light pulse has the same coordinate

representation in both coordinate systems. Along the x -axis

ctxc ±=± , (5)

and

tcxc ′±=′± . (6)

Equations (5) and (6) show Einstein's postulate in mathematical form. The (+)

and (-) signs in equations (5) and (6) indicate a rightward and leftward traveling light

pulse, respectively. Equations (1) through (6) suggest an ostensible contradiction. The

right side of the light pulse relative to B in coordinate system K seems to be traveling

more slowly than c relative to B, and the left side seems to be exceeding the speed c .

To resolve the contradiction, one must realize that vtxx −=′ , and that tt ≠′ (in

b Notice the prime on the time parameter in equation (3). Time is not necessarily the

same from one coordinate system to another. This is how the paradox is resolved.

7

contradiction to the Galilean transformation that says tt =′ , always). The next stepc is

to rewrite equations (5) and (6) as

0=−+ ctxc and 0=′−′+ tcxc

for the rightward traveling light pulse, and

0=+− ctxc and 0=′+′− tcxc

for the leftward traveling light pulse. The assumption that an affine transformation

(linear in this particular example) exists between the coordinates in K and the

coordinates in K' leads to3

)()( tcxactx cc ′−′=− ++ , (7)

and

)()( tcxbctx cc ′+′=+ −− , (8)

where a and b are arbitrary constants. This affine transformation is assumed to apply

to all events in space-time, not just those on the light cone, since a light cone can

always be defined through any given event. Generalizing +cx and −cx to x and +′cx and

−′cx to x′ , equations (7) and (8) can be rearranged to give the primed coordinates in

terms of the unprimed coordinatesd

c This step is not at all obvious. The motivation is based on the desire to find a

transformation for all points, not just those on the light cone.

d This results in two equations, four variables, and two arbitrary constants (to be

determined subsequently). So, given two of the variables, the equations may be solved

for the other two, up to the arbitrary constants.

8

( ) ( )( )

( )ctab

bababax

abbax

22+

+−

++

=′

and

( )( )

( ) ( ) tab

bacx

abba

babat

22+

++

+−

=′ .

The role of the two arbitrary constants, a and b , can be fulfilled by two different

arbitrary constants, β and γ , defined ase

( )( )ba

ba+−

=− β and ( )ab

ba2+

=γ

so that

ctxx βγγ −=′ , (9)

and

tcxt γβγ +−=′ . (10)

The arbitrary constant, β , can be found by using the primed and unprimed

coordinates to describe the position of observer B. Combining equation (9) with

equations (2) and (4) gives

cv

=β . (11)

To find the arbitrary constant, γ , symmetry must be used. Equations (9) and (10) give

the coordinates of an event as seen by observer B, in terms of the coordinates as seen

e This does not change the generality, since the number of arbitrary constants does not

change. These particular symbols are chosen in anticipation of the result and

recognition of the standard usage.

9

by observer A, given that B is moving with a velocity, v , in the x direction as seen by

observer A. The symmetry suggests that there should also be two such equations to

tell B where an event will appear in A's perspective. The coordinate values should also

match between the two sets of equations if they are to be meaningful. Equation (11) is

used to replace β in equations (9) and (10).

tvxx γγ −=′ , (12)

and

tcxvt γγ +−=′ 2 . (13)

Solving equations (12) and (13) for x and t gives

tvc

cvxvc

cx ′−

+′−

=)()( 22

2

22

2

γγ, (14)

and

tvc

cxvc

cvt ′−

+′−

=)()( 22

2

22

2

γγ. (15)

Since the velocity of A relative to K' is the opposite of the velocity of B relative to

K, the v 's must be multiplied by -1 in equations (14) and (15) in order to compare the

form to equations (12) and (13). In doing this, it is seen that all four factors agree so

that γ must satisfy

)( 22

2

vcc−

=γ

γ .

This gives

22 /1

1cv−

=γ . (16)

10

Finally, note that no transformation of the y or z coordinates occur, so that

zzyy

=′=′ . (17)

Equations (12), (13), and (17), together with the definitions (11) and (16), give the

primed coordinates of an event in space-time in terms of the unprimed coordinates.f In

matrix notation, this can be written as

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

′′′′

zyxct

zyxtc

100001000000

γβγβγγ

. (18)

The inverse of the matrix in equation (18) represents a boost in the opposite direction,

which is accomplished by changing the (-) signs to (+) signs.

2.3. Tensors

The previous section emphasized an algebraic derivation of the Lorentz boost.

However, there is a much more geometrical interpretation of the Lorentz boost. The

benefit of a geometrical interpretation is that it provides an intuitive understanding of the

effects of a Lorentz boost on the coordinates (length contraction and time dilation). To

develop an understanding of the representation of physical objects by tensors, consider

three immediate examples of three different tensor ranks: the speed of light as a scalar,

f The primed coordinates are the coordinates of the event as seen by an observer

moving with a speed v in the x direction as seen with respect to the unprimed

coordinate system. Since the orientations of the coordinate systems are arbitrary, they

can always be rotated so that their x axes coincide with the direction of the boost.

11

an event (or, more appropriately, a displacement) in space-time as a 4-vector, and the

metric as a second rank tensor.

2.3a. Scalar Fields (0th Rank Tensors)

The speed of light is a constant scalar field in special relativity. Scalar fields are

tensors of rank 0, and vice versa. Being a field means that the object is defined over a

connected region of space-time. Being a constant means that the object has the same

value at every point in the region of interest.

Scalar fields are invariant to Lorentz transformations. Consider an arbitrary point

in space-time, (also called an "event"), represented in shorthand notation by, µx . Let

an arbitrary scalar field, φ , take the value, ( )µφ x , at the point µx . Let primes indicate

transformed values under some Lorentz transformation. Since φ is a scalar field, it

satisfies the relationship

( ) ( )µα φφ xx =′′ . (19)

Equation (19) defines φ as a tensor of rank 0. This means that the value of the

scalar field at an arbitrary point in space-time transforms to the same value at the

transformed point in space-time under a Lorentz transformation. In other words,

changing the space-time perspective (coordinate system) does not change the

numerical value of the scalar field at any given point in space-time.

2.3b. Vector Fields (1st Rank Tensors)

Increasing the rank of the tensor increases the level of complexity. Equation (18)

can be written in component form as

∑=

=′3

0ν

νν

µµ xax , (20)

12

where µx′ is the thµ component of the four-component column vector on the L.H.S. of

equation (18), νx is the thν component of the four-component column vector on the

R.H.S. of equation (18), and νµa is the component in the thµ row and thν column of the

matrix in equation (18).g The scripts run from 0 to 3.h Einstein's summation convention

provides a shorthand notation for equation (20). A summation from 0 to 3 over any

Greek-letter index that appears once as a superscript and once as a subscript in a given

term is implied in Einstein's summation convention.12 Using this convention, equation

(20) can be written as12 i

νν

µµ xax =′ . (21)

This kind of summation is called "contraction." An event in space-time can be

represented by a position 4-vector, which can in turn be represented by one temporal

component (ct ), and three spatial components ( x , y , and z ). The first component of

g In other words, equation (20) is the operation of a matrix on a vector in linear algebra

written out explicitly.

h Starting the index at 0 is merely a convention. This convention will be used in this

paper since it is ubiquitous in the literature.

i There is no physical significance to what specific Greek letter is used for any given

index. The letters µ and ν happen to be popular choices. What is physically

significant is whether or not the specific letter is repeated and therefore involved in a

summation.

13

the position 4-vector,j 0x , is identified with the product of the speed of light with the time

of the event, ct . The other three components of the position 4-vector, 1x , 2x , and 3x ,

are arbitrarily identified with the Cartesian components of the position 3-vector that

represents the point in space at which the event occurs, namely, ( x , y , z ). This refers

explicitly to the νx on the R.H.S. of equation (21). The µx′ on the L.H.S. is treated

similarly, but with respect to components in the primed coordinate system.

It is more appropriate to express equation (21) in terms of differentials.k

νν

µµ dxaxd =′ , (22)

where the coordinates refer to some trajectory represented as a curve in 4-dimensional

space-time.l Equation (22) establishes a paradigm2 for a tensor of rank 1 in the context

of special relativity (Minkowski space-time)12 in terms of its components (the

components in equation (22) are the contravariant components). 4-vectors are tensors

of rank 1, and vice versa.2 The components, νµa , are the corresponding components of

j Do not confuse these upper indices with exponents. Exponentiation will be indicated

by first putting the variable in parenthesis, unless the context makes the use as an

exponent obvious.

k The reason behind this, though not extremely complicated, is deferred to the next

chapter in the discussion of curved manifolds. As a brief justification, it involves the

more appropriate concept of a tangent vector from differential geometry.

l It is chosen as a scalar so that it remains unaffected by the Lorentz transformation

matrix and is thus a global parameter.

14

the Jacobian transformation matrix. In other words, equation (22) just shows the

application of the chain rule in multivariable calculus such that

νν

µµ dx

xxxd∂′∂

=′ , (23)

where Einstein's summation convention is being used. Equation (23) shows the

geometrical meaning of the Lorentz transformations.

A tensor of rank 1 is similar to a vector in linear algebra in the sense that the

matrix multiplication of a vector has the form of equation (23). However, there is a

stronger requirement on 4-vectors in the geometrical context of tensors. The

contravariant components of a tensor of rank 1 (4-vector) must transform according to

equation (22) under a Lorentz transformation. That is, no extra terms should result from

a Lorentz transformation than those that appear in equation (22).

The components of a vector field transform under a Lorentz transformation.

Consider an arbitrary point in space-time represented in shorthand notation by, νx . Let

the contravariant components of an arbitrary vector field, A , take the values, ( )νµ xA ,

at the point νx . Let primes indicate transformed values under some particular Lorentz

transformation. Since µA are the contravariant components of a vector field, they

satisfy the relationship

( ) ( )νµµ

αβα xAaxA =′′ . (24)

Equation (24) defines A as a tensor of rank 1. This means that the value of the

contravariant components of a vector field at an arbitrary point in space-time transform

into a contraction with components of the Lorentz transformation matrix at the

transformed point in space-time under a Lorentz transformation. In a sense, the value

15

of the tensor itself does not change. However, the need to represent the tensor as a set

of 4 numbers requires a transformation of these four numbers according to equation

(24), much like rotating the coordinate axes changes the individual components of a

vector without actually changing the vector itself.

The components of a tensor can also take the covariant form. The paradigm for

this form of component is the partial derivative operator, which can be written using the

shorthand

µµ x∂∂

≡∂ . (25)

An important difference in the notation for covariant components is that the index is

written as a subscript, rather than a superscript. The transformation rule for such

components is similar to equation (24), except that the components have subscripts

instead of superscripts and the indices on the components of the transformation matrix

are switched:

( ) ( )νµα

µβα xAaxA =′′ . (26)

The two different forms of components are discussed in more detail in the next two

subsections.

2.3c. Second Rank Tensors

In order to have a sense of scale ("closeness"), a scalar productm is defined

using a tensor of rank 2 called the metric tensor. In special relativity there are four

dimensions, so a tensor of rank 2 will have 42 = 16 components, in general. These

m The scalar product is a generalization of the dot product to spaces that are not

Euclidean.

16

tensors seem quite similar to 4x4 matrices in linear algebra, but there are several

differences that make this comparison dangerously confusing. One confusing similarity

between second rank tensors and 4x4 matrices is the compact manner of listing the

components in four rows and four columns. This makes a second rank tensor look

much like a matrix, but, keep in mind that the tensor is an object independent of the

particular arrangement of components. However, out of convenience, the components

of the metric tensor in special relativity are usually given as12

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−=

1000010000100001

µνη . (27)

The scalar product is written in component form as

νµµνη vu=⋅ vu , (28)

where u and v are arbitrary 4-vectors, and µu and νv are their contravariant

components.n The scalar product is a true scalar in the sense that it is a tensor of rank

0. As an example, consider the squared distance from the origin to a point in space. In

3-dimensional Euclidean space, this can be found from the Cartesian components as

the sum of the squares of the components. This is actually the scalar (dot) product of

the position vector with itself. In flat space-time (special relativity), the distance is

generalized to something called the proper distance, but is fundamentally the same: it

is the scalar product of the position 4-vector with itself. However, in space-time this is n Don't forget that, since the Greek indices appear once as a subscript and once as a

superscript, they are summed over, from 0 to 3. This summation is called contraction

when both factors in the summation are components of tensors.

17

not exactly the sum of the squares. Using equation (28) with the definition from

equation (27), the square of the differential proper distance is given by

νµµνη dxdx=⋅dxdx

( ) ( ) ( ) ( ) =−−−=23222120 dxdxdxdx

222222 dsdzdydxdtc =−−− , (29)

where ds is the differential proper distance. Notice that the surface, 0=ds , gives a 4-

dimensional cone. This is called the light cone,o because it represents the set of all

events in space-time through which a ray of light could pass, given that it passes

through the origin of 3-dimensional space at time 0=t . If 0≠ds , then equation (29)

gives a 4-dimensional hyperboloid.p

Derived from the differential proper distance is the fundamentally important

scalar called the differential proper time, which is literally just the differential proper

distance divided by the speed of light.

( )222221 dzdydxdtcc

d ++−=τ , (30)

o It is called a "cone," but don't forget that this is a 4-D cone. Picture a sphere that

shrinks down to a point and then immediately expands again. If that point is the origin,

and the radius of the sphere shrinks and expands at the constant rate, c , then this

shrinking and expanding sphere is the light cone. If space were two dimensional, then

this could be represented as the familiar version of a cone.

p That is why the hyperbolic functions are a very straightforward way to express the

Lorentz transformations.

18

where τd is the differential proper time. This gives the time experienced by an

observer who undergoes a given displacement in space-time. This is the parameter

used to generalize the time derivative in many physical laws.

The usefulness of second rank tensors has been demonstrated by the Minkowski

metric tensor. In general, the components of a second rank tensor field follow a certain

transformation rule. Consider an arbitrary point in space-time represented in shorthand

notation by, κx . Let the contravariant components of an arbitrary second rank tensor

field, B , take the values, ( )κµν xB , at the point κx . Let primes indicate transformed

values under some particular Lorentz transformation. Since µνB are the contravariant

components of a second rank tensor field, they satisfy the relationship

( ) ( )κµνν

βµ

αλαβ xBaaxB =′′ . (31)

Equation (31) defines B as a tensor of rank 2. The contravariant components of

a second rank tensor field at an arbitrary point in space-time transform into a contraction

of each index with components of the Lorentz transformation matrix at the transformed

point in space-time under a Lorentz transformation. Again, the tensor itself does not

change, but the contravariant components change according to equation (31).

At this point one should notice that there is one Lorentz transformation factor for

every rank of the tensor. As in the case for the covariant components of a first rank

tensor, in the transformation rule for the covariant components of a second rank tensor,

the indices on the components of the Lorentz transformation matrices are switched. A

more detailed discussion of the distinction between contravariant form and covariant

form follows in the next subsection.

19

2.3d. Superscripts vs. Subscripts

There is a shorthand notation for the contraction of the metric tensor with another

tensor,q called raising and lowering an index, or "index gymnastics."2 Since the metric

tensor plays a fundamental roll in the analysis, it frequently appears in the formulae.

Rather than explicitly write it out, it is more convenient to use this shorthand. If the

metric tensor is contracted with the superscript of another tensor, then the superscript is

written as a subscript and the metric tensor is omitted. As an example, the scalar

product can be simplified using the shorthand as

µµν

ννµ

µνη dxdxdxdxdxdx == . (32)

This introduces a new kind of component, which is indexed with a subscript, called

covariant. Intuitively, the contravariant components (with the superscript) refer to the

coordinates in terms of the constant coordinate hyperplanes, while the covariant

components (with the subscript) refer to the coordinates as measured along the axes

(which are the intersections of all other hyperplanes). This gives the scalar product the

intuitive meaning of the number of hyperplanes through which a line segment passes.13

The partial derivative is the paradigm for the covariant component,2 since it holds all

other variables fixed, and thus "moves along the coordinate axis." In Cartesian

coordinates, the distinction is trivial, since the coordinate system is orthogonal. In

special relativity there is a non-trivial distinction, in that, changing between contravariant

and covariant form changes the sign on the 1st, 2nd, and 3rd components. This can be

q This is actually more than just a convenient shorthand; it shows that the contraction of

the components of the metric tensor with the contravariant components of a 4-vector

gives the covariant components of the 4-vector.

20

seen by comparing equation (29) with equation (32). Since the partial derivative is the

paradigm for the covariant component, it is given a special symbol.

fxf

µµ ∂≡∂∂ , (33)

where f is a scalar function of the position in space-time. Notice that the symbol takes

a subscript, indicating that it is covariant. So, the differential is the paradigm for the

contravariant form of a vector, and the partial derivative is the paradigm for the

covariant form of a vector.

The components of the metric tensor also have a contravariant form that raises

the index of covariant components to make them contravariant. The different types of

components of the metric tensor satisfy the relationship:12

νµ

ανµα δηη = , (34)

where νµδ is the Kronecker Delta.r Notice that the two forms of the components of the

metric tensor are contracted on one index, leaving one free indexs from each tensor. In

general, the contraction takes away one rank from each tensor, so, the components of

two 2nd rank tensors contracted on one of their indices leaves 2 - 1 + 2 -1 = 2 indices,

and therefore results in components of a 2nd rank tensor. The contraction of the

components of two 1st rank tensors (4-vectors) results in a tensor of rank 0 (scalar).

Also notice that the result preserves the placement of the remaining indices as a

subscript or superscript. It turns out that the covariant components of the Minkowski

r The Kronecker Delta can be thought of as the mixed components of the metric tensor

of space-time.

s A free index is an index that is not involved in the contraction.

21

metric tensor are the same as the contravariant components. So, equation (23) also

represents the covariant components.

There is another 2nd rank tensor, called the Faraday tensor, which encapsulates

the electric and magnetic field into one physical object. It is developed from first

principles of electrostatics in the next three sections.

2.4. Generalizing Electrostatics to the Time Dependent Gauss'

Law and the Ampere Maxwell Law

Here, a brief review of electrostatics is given. The goal is to propose the

minimum number of physical postulates in order to generalize to a relativistic version of

the electromagnetic field. This generalization results in a second rank tensor, called the

electromagnetic field strength tensor, or Faraday tensor. Maxwell's equations follow as

a natural consequence of this generalization.

Coulomb's law states that the force between two charges is proportional to the

product of the charges and inversely proportional to the square of the distance between

them. For two charges, q and q′ , located at points in space represented by the

position vectors, xr and x′r , respectively, Coulomb's law gives the force, Fr

, experienced

by the charge, q , as

304 xx

xxqqF rr

rrr

′−

′−′=

πε, (35)

where 0ε is the permittivity of free space. The electric field, Er

, at the point in space

represented by the position vector, xr , due to charge, q′ , is defined as

qFEr

r= . (36)

22

If the source charge, q′ , is generalized to the charge density at a point in space,

xr′ , as a function of the position vector that represents that point, )(3 xxdqd r′′=′ ρ , where

xd ′3 is a small volume element, then, using the principle of superposition, the electric

field, ( )xE rr, is given as an integral over all of space.

( ) ∫′ ′−

′−′′=V xx

xxxxdxE 33

0

)(4

1rr

rrrrr

ρπε

. (37)

The divergence and curl of the electric field can be found by taking the

divergence and curl of the integral in equation (37). As a resultt

( ) )(1

0

xxE rrrρ

ε=⋅∇ , (38)

and,

( ) 0=×∇ xE rr. (39)

Equations (38) and (39) are the equations of electrostatics that follow from

Coulomb's Law. The electric charge density, )(xrρ , is the 0th contravariant component

of a 4-vector.u,14 This 4-vector is called the charge-current density 4-vector.14 The

contravariant components of the charge-current density 4-vector are:14 cj ρ=0 ,

xjj =1 , yjj =2 , and zjj =3 . Notice the similarity to the assignment of the components

in the position 4-vector. This indicates that the electric charge density transforms in the

same way that time transforms under a Lorentz transformation.

t The derivation for equations (38) and (39) can be found in Appendix A.

u The justification for the static charge being the 0th component of a 4-vector is shown in

Appendix B.

23

Consider the contraction of the partial derivative 4-vector with the charge-current

density 4-vector, µµ j∂ . This is called the 4-divergence of the charge-current density.

From Section 3.d, this contraction can be recognized as a scalar field. Consider a static

distribution of charge in its own rest frame, so that 0=ij . In this case the scalar field is

just the time derivative of the charge density. Since charge is a conserved quantity,

then the time rate of change of the static charge distribution is

( )( ) 00

0

=∂∂

=∂∂

=∂∂

xj

tcc

tρρ (40)

in its own rest frame. Since the contraction is a scalar field, then in any coordinate

system

0=∂ µµ j . (41)

Equation (41) is the continuity equation,

jt

r⋅−∇=

∂∂ρ , (42)

where jr

is the 3-dimensional electric current density vector.v

Replacing the charge density, ρ , in equation (38) with the 0th component of the

charge-current density 4-vector,

( ) ( )xjc

xE rrr0

0

1ε

=⋅∇ . (43)

v Physically, equation (42) says that, if there is an increase or decrease of charge at a

point in space, then there is a net current flow into or out of that region of space that

contains the charge.

24

Since the R.H.S. of equation (43) transforms as the 0th component of a 4-vector,

so must the L.H.S. The divergence of the electric field generalizes to a contraction of

the partial derivatives with the contravariant components of a tensor. This, along with

the 4-vector transformation property of the L.H.S., requires that the components of the

electric field be generalized to components of a 2nd rank tensor.14

30

20

10

FE

FE

FE

z

y

x

=

=

=

(44)

so that

000

0 FFE ∂−∂=⋅∇ µµ

r. (45)

These new components that have been introduced are components of the

Faraday tensor. Again, since equation (45) must transform as the 0th component of a 4-

vector, the last term, 000F∂− , must vanish.w Equation (43) then becomes

0

0

0 1 jc

Fε

µµ =∂ . (46)

Substituting the definitions that have been used so far in this section, equation

(46) is a generalization of equation (38) to a time-dependentx electric field and current

wThe last term in equation (45) must vanish because it transforms as the 000 mixed

component of a 3rd rank tensor, and therefore does not transform as the 0th component

of a 4-vector.

x The time dependence is a consequence of the requirement that the physical objects

must be tensor fields, which must be defined as functions of space-time, not just

functions of space, in order to be Lorentz invariant.

25

density. In order to have form invariance under Lorentz transformations, the other three

components of the 4-vector must be included. Therefore, equation (46) generalizes to14

νµνµ ε

jc

F0

1=∂ . (47)

There is only one subtle difference in the notation between equation (46) and (47),

namely, that the 0 superscript has become a Greek letter ν . Physically, however, there

is a very profound difference, namely, that equation (47) accounts for all components of

the Faraday tensor whereas equation (46) does not. A consequence of this natural

generalization is that equation (47) gives another one of Maxwell's laws, the Ampere-

Maxwell law.14

To find the other components of the Faraday tensor, the divergence of equation

(47) gives14

0=∂∂ µνµν F , (48)

by equation (41). Equation (48) suggests that the Faraday tensor is antisymmetric.y,14

Assuming that the Faraday tensor is antisymmetric reveals 7 more of its components by

symmetry: 0=µµF , xEF −=01 , yEF −=02 , and zEF −=03 . To find the remaining 6

components, equation (47) is examined. As previously mentioned, the ν = 0

component of equation (47) returns Gauss' Law for a time-dependent electric field and

charge density.z

y For a discussion of the antisymmetry of the Faraday tensor, refer to Appendix C.

z The result of Gauss' Law with time dependence should not be at all surprising

considering the development up to equation (46).

26

( ) ( )txtxE ,1,0

rrrρ

ε=⋅∇ , (49)

where the argument represents a dependence on space as well as time. For ν = 1,

equation (47) givesaa

t

Ej

cF

zcF

yc x

x ∂∂

+=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

0

31212 1

ε, (50)

and similarly for the ν = 2 and 3 terms. Equation (50) gives the x -component of the

Ampere-Maxwell Law,14 given certain assignments for the components of the Faraday

tensor that appear, namely xcBF −=23 , ycBF −=31 , zcBF −=12 , xcBF =32 , ycBF =13 ,

and zcBF =21 . Therefore, the Faraday tensor may now be sumarized in a 4x4 matrix in

terms of the electric and magnetic fields as

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

−−−−

=

00

00

xyz

xzy

yzx

zyx

cBcBEcBcBE

cBcBEEEE

F µν , (51)

where the first index gives the row, starting with zero from the top, and the second index

gives the column, starting with zero from the left. If these assignments are made, then

equation (50) gives the Ampere-Maxwell Law

( ) ( ) ( )t

txEc

txjtxB∂

∂+=×∇

,1,, 20

rrrrrr

µ , (52)

where 0µ is the permeability of free space.

This accounts for all of the components of the Faraday tensor. Finding the last

six is a bit speculative. Reference to the Ampere-Maxwell Law seems to break the

aa For the details of the derivation of equation (50), refer to Appendix D.

27

promise that only electrostatics would be needed. The resolution is that the naming of

the components is arbitrary. The oblique components of the Faraday tensor were

recognized to satisfy the Ampere-Maxwell equation from equation (50), but the tensor

had already been defined physically to satisfy equation (47), which describes the

fundamental response of the Faraday tensor to the charge-current density. In other

words, equation (47) shows that these oblique components satisfy the Ampere-Maxwell

equation, and therefore they are associated with the magnetic field components. The

naming convention should not be obscured as a physical indication; equation (47) gives

the Ampere-Maxwell equation, not vice versa. For the reader who is still not convinced,

ultimately, the definition for these oblique components is verified operationally by their

appearance in the Lorentz force law in Section 6.

2.5. Generalizing Electrostatics to Faraday's Law and the

Law of No Magnetic Monopoles

Faraday's Law, equation (39), for electrostatics was obtained in Section 4 from

the basic principles already mentioned. Written in component form

03

1

3

1

=∂∂∑∑

= =k j j

kijk x

Eε , (53)

where ijkε is the Levi-Civita symbol defined as14

⎪⎩

⎪⎨

⎧−+

=3,2,10

3,2,113,2,11

ofnpermutatioanotisijkifofnpermutatiooddanisijkifofnpermutatioevenanisijkif

ijkε . (54)

Using the notation from section 3 and equation (44), equation (53) becomes

03

1

3

1

0 =∂∑∑= =k j

kjijk Fε . (55)

28

The Levi-Civita symbol can be generalized to a 4th rank tensor called the Levi-

Civita tensor,14 by using a definition similar to equation (54) for the contravariant

components of the tensor.

⎪⎩

⎪⎨

⎧−+

=3,2,1,00

3,2,1,013,2,1,01

ofnpermutatioanotisifofnpermutatiooddanisifofnpermutatioevenanisif

κλµνκλµνκλµν

εκλµν . (56)

Replacing the Levi-Civita symbol in equation (55) with the appropriate

contravariant components of the Levi-Civita tensor, and lowering the indices on the

components of the Faraday tensor,

03

1

3

10

0 =∂∑∑= =k j

kjijk Fε , (57)

where14 ijkijk εε −=0 , and 0

00k

kk FFF −== µννµηη . Recognizing that the component of

the Levi-Civita tensor vanishes if any of the other indices = 0, equation (57) is identical

to

000 =∂ αµ

µναε F . (58)

To generalize this to a tensor equation, the 0 must be generalized to an index

that ranges from 0 to 3. Since the components of the Levi-Civita tensor are constant,

they can be moved between the partial derivative to directly multiply the components of

the Faraday tensor. Equation (58) can also be multiplied by 1/2. This gives

( ) 021 =∂ αβ

µναβµ ε F . (59)

The quantity in parenthesis is defined as µνF∗ , the dual of the µνF component of the

Faraday tensor.14 These components can be expressed as a matrix similar to equation

(51).

29

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

−−−−

=∗

00

00

xyz

xzy

yzx

zyx

EEcBEEcBEEcBcBcBcB

F µν . (60)

Replacing the quantity in parenthesis using this definition, equation (59) becomes:

0=∂ ∗ µνµ F . (61)

"A generalization of Helmholtz's theorem states that an antisymmetric second-rank

tensor is completely determined by specifying its divergence and the divergence of its

dual."15 These specifications have been made by equations (47) and (61) in terms of

the charge-current density 4-vector.

For ν = 0, using the definition of the dual, and the suggested antisymmetry of the

Faraday tensor from Section 4, equation (61) gives

0111 211332

=∂∂

+∂∂

+∂∂

zF

cyF

cxF

c, (62)

where the factor of c/1 has been introduced arbitrarily without changing the validity of

the equation. Equation (62) gives the law of no magnetic monopoles using the

component definitions in equation (51).

0=⋅∇ Br

. (63)

This supports the discussion in the previous section concerning the relationship of these

components to the magnetic field.

For ν = 1, equation (61) gives:

0313

212

111

010 =∂+∂+∂+∂ ∗∗∗∗ FFFF , (64)

30

and similarly for ν = 2 and 3. Rearranging equation (64) and making the assignments

to the components of the Faraday tensor according to equation (51) gives Faraday's

Law for time-dependent electric and magnetic fields.

( ) ( )t

txBtxE∂

∂−=×∇

,,rr

rr. (65)

This further supports the discussion in the previous section concerning the relationship

of these components to the magnetic field.

2.6. The Lorentz Force

The previous sections developed the Faraday tensor from a few reasonable

assumptions. As a result, two similar equations were found, (47) and (61), that

determine the Faraday tensor in terms of a given charge-current density 4-vector. In

this section, the response of a point-charge is found to a given Faraday tensor, which

completes the mathematical description of the interaction between charge and field. To

begin the development, Newton's second law is considered in the context of the electric

field (equation (36)).

Eqdtpd rr

= , (66)

where pr is the 3-dimensional momentum vector of the charged point-particle of charge

q , and Er

is the electrostatic field at the charge. Equation (66) is a first order, linear,

ordinary differential equation. Applying the usual technique of separating the dependent

variables from the independent variables, an implicit equation can be obtained in the

derivatives of pr and t with respect to the proper time τ .

Eqddt

dpd rr

ττ= . (67)

31

As in the previous 2 sections, the components of the electric field are identified with

components of the Faraday tensor 10FEx → , etc., according to equation (44). Applying

this direct substitution to equation (67) gives

0ii

Fqddt

ddp

ττ= , (68)

where ip is the thi component of the momentum vector pr . Multiplying the R.H.S. of

equation (68) by cc gives

00

000 iiii

FucqF

ddx

cqF

ddtc

cq

ddp

=⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

τττ, (69)

where 0u is the 0th covariant component of the proper velocity νu defined as the proper

time derivative of the position 4-vector.

τν

ν ddx

u ≡ . (70)

The L.H.S. of equation (69) can be generalized to a 1st rank tensor in a

straightforward manner by changing the Latin superscript i to a Greek letter µ , thus

including all four components. This will also change the i on the R.H.S. of the equation

to a µ . Since the L.H.S. is a 1st rank tensor, so must be the R.H.S. This requires that

the 0 subscript and superscript be generalized to contracted indices rather than free

indices.

µνν

µ

τFu

cq

ddp

= . (71)

Equation (71) is the relativistic equation of motion for a charged particle. It includes

every component of the momentum 4-vector µp (and velocity 4-vector νu ) of the

32

charged particle and every component of the Faraday tensor µνF . Consider, as an

example, µ = 1, which represents a typical spatial component (the x -component). This

gives, directly from equation (71),

133

122

100

1

FucqFu

cqFu

cq

ddp

++=τ

. (72)

Using the chain rule and equation (70), and substituting xpp =1 ,

133122100 Fddx

cqF

ddx

cqF

ddx

cq

dtdp

ddt x

ττττ++= . (73)

The components of the position 4-vector can be identified according to section 2.3b,

and the 10F component of the Faraday tensor is the x -component of the electric field.

The chain rule can also be applied to the R.H.S. of the equation.

⎥⎦⎤

⎢⎣⎡ −−+= 1312 F

dtdzF

dtdy

ddt

cqE

dtdtc

ddt

cq

dtdp

ddt

xx

τττ. (74)

Simplifying,

[ ]1312 FvFvcqEq

dtdp

zyxx −−+= , (75)

where dtdyvy = and

dtdzvz = . Letting zBcF −=12 and yBcF =13 in accordance with the

previous 2 sections, equation (75) gives the x -component of the force experienced by a

point particle of charge q moving with velocity vr through an electric field Er

and

magnetic field Br

.

[ ]xxx BvqEq

dtdp rr

×+= . (76)

The y and z -components follow similarly.

33

The µ = 0 component of equation (71) yields a well-known feature of

electromagnetism, and does so naturally from the principle of relativity.

⎟⎠⎞

⎜⎝⎛ ++=⎟

⎠⎞

⎜⎝⎛ ++= zyx E

dtdzE

dtdyE

dtdx

cqF

ddx

FddxF

ddx

dtd

cq

dtdp 033022011

0

ττττ , (77)

which can be rearranged to give

( ) ( ) ( )Evqdt

energyddt

cpd rr⋅=≡

0

. (78)

Equation (78) gives the power delivered to the charge by the electric field, and shows

that the magnetic field does no work.

2.7. Conclusion

Assuming that Coulomb's Law and the principle of superposition are valid for a

static electric field and static charge distribution, the electrostatic field is found to be

conservative and Gauss' Law obtained. By assuming that charge is a conserved scalar,

the equation of continuity was found for the charge density. This allowed Gauss' Law

for the electrostatic field to be generalized, using the principle special relativity, to

Gauss' Law for electrodynamics and the Ampere-Maxwell equation. The curl of the

electrostatic field was also generalized to give the law of no magnetic monopoles and

Faraday's Law. Finally, the force of the electrostatic field was generalized to the

Lorentz force and an equation for the rate of energy equal to the power delivered to a

point charge in an electric field.

All of this information can be written in terms of three tensor equations in special

relativity, (47), (61), and (68), that maintain their form under all Lorentz transformations.

The first two equations determine the response of the field to charged matter, which is

represented by the response of the Faraday tensor to the charge-current density 4-

34

vector. These equations are Maxwell's equations. The third equation determines the

response of charged matter to the field, which is represented by the response of a point

charge to the Faraday tensor. This equation is Newton's second law of motion with the

Lorentz force. The mutual interaction between the field and charge upholds Newton's

third law of motion.

35

CHAPTER 3

GENERAL RELATIVITY: EINSTEIN'S GRAVITATIONAL FIELD EQUATION FROM

NEWTON'S LAW OF UNIVERSAL GRAVIATION

3.1. Introduction

After completing his theory of special relativity, Albert Einstein realized that the

next great challenge would be to make gravitation consistent with the special theory of

relativity.2 Isaac Newton himself found fault in his universal law of gravitation because it

required a "spooky action at a distance."16 In order to have an instantaneous action at a

distance, an interaction must travel at an infinite speed. This becomes a problem for

causality in special relativity, because, what is simultaneous in one frame of reference is

not simultaneous in a boosted frame. In other words, an action in one frame of

reference could inconsistently be a re-action in another frame of reference. The

alternative to instantaneous action at a distance is an interaction that propagates at the

speed of light, which is accomplished in general relativity. Another problem with

Newton's universal law of gravitation is the source term, namely the mass. Obviously,

what is a static mass distribution in one frame of reference induces a mass current in a

boosted frame. So, the source of gravitation must demonstrate this in general relativity

as a tensor. It turns out that the source of gravitation is matter and energy, partly as a

consequence of Einstein's equivalence principle, so a second rank tensor is used, in

contrast to the source of the electromagnetic field, which is a first rank tensor.2 Another

consequence of the equivalence principle, and perhaps the most profound (certainly the

most popular) feature of general relativity, is that the gravitational field is actually a

distortion of space-time itself.2

36

There are two basic approaches to demonstrate the correspondence between

Einstein's gravitational field equation and Newton's universal law of gravitation. One of

these is to show that Einstein's gravitational field equation reduces to Poisson's

equation for gravity in the limit of static, weak fields. This is mathematically

straightforward, and involves neglecting certain terms in the equation at the appropriate

stages. Newton's universal law of gravitation has been demonstrated as a limiting form

of Einstein's equation by several authors.17,18,19 The other approach is to show that

Einstein's gravitational field equation is an inevitable consequence of Newton's

universal law of gravitation when certain assumptions are made concerning the physical

universe. The latter approach is taken in this chapter.

Many other authors have shown Einstein's gravitational field equation as a

generalization of Newton's universal law of gravitation. Ohanian shows that Einstein's

gravitational field equation is the result of a generalization of Poisson's equation, with

the assumption that the gravitational potential is the 00 component of the space-time

metric tensor.20 Misner, et. al. gives six "reasonable axiomatic structures" that lead to

Einstein's equation, with the underlying assertion that the generalization must lead to

Einstein's equation.21 Kobe shows a heuristic treatment of the speed of light to

establish a relationship between the gravitational potential and the metric tensor, similar

to Einstein's first theory using the speed of light as a scalar field.22,23 Costa de

Beauregard introduces a "generalized Poisson potential" that is not closely related to

the gravitational potential in Newton's universal law of gravitation.24 Moore derives

Schwarzschild's equation from Poisson's equation, but does not use tensors (general

covariance) nor develops the general form of Einstein's equation.25 Mannheim

37

mentions the relationship between Einstein's gravitational field equation and Poisson's

equation, but does not give the derivation.26 Rindler proposes a canonical form for the

metric of curved space-time, compares Poisson's equation to the geodesic equation for

curved space-time and then relates the gravitational potential to the Christoffel

symbol.27 Bondi also assumes a canonical metric for a spherically symmetric mass

distribution and does not give detailed justification for the use of the stress-energy

tensor or Einstein tensor, but does give an excellent intuitive illustration of non-

Euclidean space.28 In a lecture on general relativity and astrophysics delivered at the

DPG School, a linear relationship is assumed between the stress-energy tensor, Ricci

tensor, and curvature scalar, in order to reflect the linear relationship of the tidal force to

the mass density in Newton's theory.29 Weinberg gives similar points in the

development that is found in this paper.30 Perhaps the most significant approach to the

one presented in this paper was by Chandrasekhar, who gives more detail in some

parts of the generalization and far less detail in others, and uses a different set of

assumptions.31

In this chapter, the concept of general covariance (tensor analysis in the context

of general coordinate transformations, in contrast to the linear transformations in special

relativity) is applied to the classical Poisson's equation for gravity, which is based on

Newton's universal law of gravitation and the principle of superposition. Poisson's

equation for gravity can then be written as the 00 component (time-like component) of a

second rank tensor equation.2 Einstein's equation is obtained by relating the mass

density (times c2) to the 00 component (being the energy component) and trace of the

energy-stress tensor.2 Poisson's equation, which is valid for slowly moving particles in

38

static and weak gravitational fields, is generalized to give Einstein's gravitational field

equation, which is valid for particles at any speed in an arbitrary gravitational field.2

A brief development of Poisson's equation for Newtonian gravity is presented in

Sec. 2. Section 3 gives a discussion of space-time curvature and introduces some

important tensors. The principle of least action for gravity is shown to be equivalent to

the definition of a geodesic in Sec. 4. In Sec. 5, the mass density (times c2) is

generalized to the stress-energy tensor, and the Bianchi identity is used to give

Einstein's equation. Section 6 gives the conclusion.

3.2. Poisson's Equation for Gravity

In 1686, Isaac Newton presented his law of universal gravitation to the world.5

The key features of this law are that the force is proportional to the product of the point

masses, and that the force is inversely proportional to the square of the distance

between the point masses. Given two masses, m and m′ , the force ( )xF rr on mass m

at xr by the mass m′ at the origin is given by Newton's universal law of gravitation:

( ) 3xxmGmxF r

rrr

′−= , (79)

where G is Newton's universal gravitational constant.5 Equation (79) can be

generalized so that m′ is at some point x′r :

( ) 3xxxxmGmxF′−

′−′−= rr

rrrr

. (80)

The forces on m at xr due to masses im at ixr ( Ni K,2,1= ) obey the principle of

superposition, so that the total force on mass m at xr is:

( ) ∑−

−−=

i i

ii

xxxx

mGmxF 3rr

rrrr

. (81)

39

If the constellation of masses in equation (81) is generalized to a continuous distribution

of mass with a mass density ( )x′rρ at the position x′r , then the force on mass m at xr

by the continuous mass distribution is given by:

( ) ( )∫ ′−

′−′′−= 33

xxxxxxdGmxF rr

rrrrr

ρ , (82)

where the integral is a volume integral on x′r over all space.

A gravitational field ( )xg rr at the displacement xr can be defined as:

( ) ( )mxFxgrr

rr≡ . (83)

Combining equations (82) and (83), the gravitational field at the point xr due to a

continuous distribution of mass with a mass density ( )x′rρ as a function of the position

x′r is given as:

( ) ( )∫ ′−

′−′′−= 33

xxxxxxdGxg rr

rrrrr ρ . (84)

The gravitational field is conservative because ( ) 0=×∇ xg rr .bb Therefore a

gravitational potential function ( )xrΦ for the gravitational field ( )xg rr can be found, which

is given by:cc

( ) ( )∫ ′−

′′−=Φ

xxxxdGx rr

rr ρ3 . (85)

bb This is shown in Appendix E

cc The derivations of equation (85) from equation (84) and equation (86) from equation

(85) can be found in Appendix E.

40

Taking the Laplacian of equation (85) gives Poisson's equation for the Newtonian

gravitational potential:

( ) ( )xGx rr ρπ42 =Φ∇ . (86)

3.3. Space-time Curvature

The Minkowski space-time of special relativity is often referred to as "flat" space-

time. The space-time of general relativity is often referred to as "curved" space-time.2

This section demonstrates the significance of this curvature and provides a mechanism

to quantify it.

Euclidean geometry has several features, called axioms, such as the existence

of parallel lines and an open set of points, that serve the same purpose as intuitive

geometric notions. The advent of Riemannian geometry demonstrated that the axioms

of Euclidean geometry are nontrivial. The most obviously nontrivial notion is the

Euclidean construct of parallel lines, which require a more careful definition in

Riemannian geometry. Riemannian geometry is the generalization of Euclidean

geometry to include the notion of curvature.

In this section the relevant distinctions between Euclidean space, Minkowski

space-time, and pseudo-Riemanniandd,32 space-time are demonstrated, and the tensors

used to describe this curvature in the context of space-time (4-D) are developed.

dd The geometry of general relativity is not truly Riemannian because the geometry of

special relativity is not truly Euclidean. This subtly is not important for the development

in this paper, but, to be rigorous, the prefix "pseudo-" is added to indicate this

distinction.

41

A key feature of Euclidean geometry is Pythagoras' theorem that essentially

defines the distance between two points to be the distance that one would measure if

one were to put a physical ruler across them. This implies that the metric tensor of

Euclidean geometry is the unit second rank tensor, the Kronecker Delta, mnδ , in

Cartesian coordinates. In special relativity, the idea of distance is generalized to proper

distance or proper time. This implies that the metric tensoree used in the geometry of

special relativity is the Minkowski metric tensor,12 µνη , in Cartesian coordinates. The

Minkowski metric tensor is generalized to the metric tensor for curved space-time,33

µνg , in general relativity. The metric tensor itself is a dynamical variable in general

relativity.34 It interacts with mass and energy and reacts accordingly. This profound

consequence of general relativity has evidence in Einstein's equivalence principle in

which he proposed gravitational and inertial mass to be equivalent.

The notion of parallel is more appropriately applied to vectors than to lines in

general relativity. Two vectors at the same point in space-time are parallel to each

other if and only if one is a scalar multiple of the other. In order to determine whether

vectors at two different points in space-time are parallel, a connection must be

established. The connection used in general relativity is parallel transport. Parallel

transport is the transporting of a tangent vector from one point to another along a

piecewise geodesic path while maintaining the orientation of the vector to each

ee A discussion of tensors can be found in Chapter 2, Section 3.

42

geodesic in each piece.35 A geodesic is a path that extremizes the proper distance or

proper time between two given points.ff,32

Parallel transport can have a peculiar consequence in a curved geometry. The

vector that is parallel transported along one path can disagree with the same vector

transported along a different path between the same two points. The simplest example

of this disagreement occurs on the surface of the Earth. Consider an arrow pointing

westward on the equator. Parallel transporting this arrow along the equator to the

opposite side of the Earth will result in a westward pointing arrow on the opposite side

of the Earth. Parallel transporting this arrow along a line of longitude to the same point

will result in an eastward pointing arrow at that point. This disagreement is a direct

result of the curvature of the geometry, and it can be used to quantify the curvature.

First, note that parallel transportation along a single geodesic is closely related to

partial differentiation.gg Next, note that the tensor product of the partial derivative tensor

with a 4-vector results in an object that is not a tensor.hh A new kind of derivative can

ff Actually, more generally, a geodesic is a curve of stationary proper length in pseudo-

Riemannian geometry. However, it is of maximum proper length for all massive

particles. A geodesic represents the "straightest" path connecting two points in curved

space.

gg Recall that a partial derivative is a change in a function with respect to a change

along a coordinate axis.

hh Appendix F shows why the tensor product of the partial derivative with a 4-vector

results in an object that is not a tensor.

43

be defined that is related to the partial derivative, traditionally called the covariant

derivative, that is a tensor.

Some new notation is introduced to simplify the expression for a partial

derivative. A comma before a lower index will be used to indicate a partial derivative

with respect to the coordinate that takes that index:

µββ

µ AA ∂≡, , (87)

where ββ x∂∂≡∂ / according to equation (25) or (33) in Section 3 of Chapter 2.

For an arbitrary 4-vector µA , the covariant derivative with respect to βx is

denoted by a semicolon before a lower index and is defined as:36

αµαβ

µββ

µ AAA Γ+∂≡; , (88)

where µαβΓ is known as a connection coefficient. In general relativity, the connection

coefficient is called the Christoffel symbol. The Christoffel symbol can be expressed in

terms of the metric tensor as:37

( )ναβαβνβναµνµ

αβ ,,, gggg −+=Γ . (89)

The term on the left-hand side of equation (88) is a second rank mixed tensor, whereas

the object in equation (87) is not. Though the covariant derivative is a tensor, two

consecutive covariant derivatives of a 4-vector do not commute, in general. This

property can be used to quantify the curvature of space-time. A fourth rank tensor,

known as the Riemann curvature tensor βµναR , is defined in terms of the Christoffel

symbols as:35

ασν

σβµ

ασµ

σβν

αµβν

ανβµβµν

α ΓΓ−ΓΓ+Γ+Γ−≡ ,,R . (90)

44

The Riemann tensor is essentially the commutator of two covariant derivatives

acting on a 4-vector.ii It indicates the path dependence of differentiation in curved

space. If the space is flat, then the Riemann tensor vanishes.

Two other important tensors come directly from the Riemann curvature tensor.

The Ricci tensor µνR is defined as the Riemann curvature tensor contracted on its first

and last indices:38

µναα

µν RR ≡ . (91)

The Riemann curvature scalar R is defined by the contraction of the two remaining

indices:38

µνααµν

µνµν

µµ RgRgRR ==≡ . (92)

The Ricci tensor and Riemann curvature scalar together satisfy the Bianchi identity:38

( ) 0;21 =− µν

µν

µ δ RR . (93)

Equation (93) is valid geometrically, meaning it is independent of the details of

the physical situation.

3.4. Principle of Least Action

Points in 3-D space extrude into 4-D space-time as worldlines. In Minkowski

space-time, the notion of a straight line is intuitively the same as that in familiar

Euclidean geometry. That is, the worldlines of free particles "look" straight in Minkowski

space-time. More formally, a geodesic is the generalization of a straight line to curved

ii Note that the left-hand side of equation (90) is a tensor, even though every term on the

right-hand side is not a tensor in itself.

45

space, and is defined as the path that maximizes the proper time experienced by a

particle that travels between two given points.39

max=∫geodesic

dτ . (94)

A geodesic is a path over which a particle will not experience any force. Any

deviation from this path will result in an experienced acceleration and a decrease in the

proper time experienced by the particle.jj Since a deviation from this path results in a

force, it should not be surprising that this requirement is related to the principle of least

action for the gravitational potential energy.

min=∫pathactual

dtL , (95)

where L is the Lagrangian function for a massive particle in a gravitational potential.

Using the definition of the Lagrangian as

VTL −≡ , (96)

equation (95) can be rewritten as

( ) min221 =Φ−∫ dtmmv

pathactual

. (97)

In the limit

µνµν η→g , (98)

equation (94) can be rewritten askk

jj The fact that acceleration reduces the experienced proper time is valid in special

relativity. This is one way to address the infamous twins paradox.

kk See Appendix F for a derivation of equation (99).

46

( )( ) min221

002

212

21 =+−∫ dtcgcmmv

geodesic

, (99)

In equation (97), gravity is treated as a force for which there exists a potential

function. This is the notion of gravity given by Newton. In equation (99), the influence

of geometry on the trajectory of an otherwise free particle is demonstrated. Solely

under the influence of gravity, the actual path of a physical particle is a geodesic.

Comparing equations (97) and (99), the gravitational potential can be expressed in

terms of the metric tensor as40

221

002

21 cgc +=Φ . (100)

Equation (100) gives the relationship between the Newtonian gravitational

potential Φ and the 00 component of the metric tensor 00g .

3.5. Einstein's Equation

The mass density, ρ , is related to the 00 mixed component of the stress-energy

tensor 00T in the cloud of dust model41,ll which Einstein used in his paper, The Meaning

of Relativity.42 First, consider the trace of the stress-energy tensor.

2~~ cuuTT ρρ µµ

µµ ==≡ , (101)

where ρ~ is the proper mass density scalar field. In the static limit, the space-like

components of the stress-energy tensor (in the cloud of dust model) vanish.

0000 =++=kkT . (102)

ll See Appendix H for a description of the stress-energy tensor.

47

This implies (numerically) that

220

0 ~ ccTT ρρ === (103)

in the static limit.

According to equation (103), the Newtonian mass density (times 2c ) can be

written as either the 00 mixed component or the trace of the stress-energy tensor. More

generally, it can be written as a linear combination of the two.

( ) 00

002 1 δρ TaaTc −+= , (104)

where a is a constant to be determined and 00δ is the 00 component of the Kronecker

Delta tensor (and therefore equal to unity). Though seemingly superfluous at this point,

the Kronecker Delta tensor is necessary for subsequent generalization in order to

maintain the rank and form of the tensor components. Substituting equation (100) and

(104) into Poisson's equation (86) gives

( )( )00

00

4002 18 δπ TaaT

cGg −+=∇ . (105)

The L.H.S. of equation (105) can be related to the 00 component of the Ricci

tensor.2,mm

00

002 2Rg −=∇ . (106)

Combining equations (105) and (106) gives

( )( )00

00

400 14 δπ TaaT

cGR −+−= . (107)

Both the L.H.S. and R.H.S. of equation (107) are 00 mixed components of

second rank tensors. Generalization of equation (107) to be form invariant under

mm See Appendix I for the generalization of 00

2 g∇ to 002 R− .

48

general coordinate transformations is simply a matter of recognizing that equation (107)

is the 00 component of a tensor equation, and that all sixteen mixed components of the

second rank tensor equation must exist in general.2

( )( )νµ

νµ

νµ δπ TaaT

cGR −+−= 144 (108)

for 3,2,1,0, =νµ .

The Riemann curvature scalar can be related to the trace of the stress-energy

tensor by taking the trace of equation (108).

( ) ( )( )4144 TaaT

cGR −+−=

π . (109)

Solving this equation for T, substituting this back into equation (108) and rearranging

gives

( )( ) ν

µν

µν

µ πδ Tc

aGRa

aR 4

434

1−=

−−

− . (110)

Energy and momentum are conserved if the 4-divergence of the stress-energy

tensor vanishes as20

0; =µνµT . (111)

On physical grounds, it is assumed that energy and momentum are conserved in the

theory. Taking the 4-divergence of equation (110) gives

( )( ) 0

341

;

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−µ

νµ

νµ δR

aaR . (112)

Comparing equation (112) with the Bianchi identity equation (93) requires

( )( ) 2

134

1=

−−

aa . (113)

Therefore the value of a is 2. Substituting 2=a into equation (110) gives

49

νµ

νµ

νµ πδ T

cGRR 42

1 8−=− . (114)

Equation (114) is Einstein's equation for the gravitational field in terms of the

Ricci tensor, Riemann curvature scalar, and the stress-energy tensor.20 Another form of

Einstein's equation is obtained by raising the lower index in equation (114).

µνµνµν π Tc

GgRR 421 8

−=− . (115)

3.6. Conclusion

Poisson's equation for gravity is a direct consequence of Newton's law of gravity

if superposition, continuity, and conservation of energy are assumed. This assumption

is valid in static, weak gravitational fields. However, in general relativity, the

gravitational field is related to the curvature of space-time. Therefore, the metric tensor

of special relativity must be generalized to the metric tensor of curved space-time.

Using the principle of least action to relate the gravitational potential to the 00

component of the metric tensor, Einstein's equation follows as a natural consequence of

tensor analysis, the Bianchi identity and the conservation of energy and momentum.

50

CHAPTER 4

CONCLUSION

Classical physical theory effectively began when Isaac Newton proposed his

classical dynamics and universal law of gravitation. This sparked a classical world view

of physical phenomena to which science remains loyal even today, in the appropriate

limits of consideration, usually referred to as "everyday experience."23 Classical

mechanics was not successfully contested until Albert Einstein introduced his special

theory of relativity. However, Einstein further extended the classical view by discovering

the appropriate way in which to generalize Newtonian gravitation.2 In this way,

Einstein's general relativity marks the completion of the classical deterministic physical

theory of the universe.2

51

APPENDIX A

Derivation of Equations (38) and (39) from Coulombs Law and

the Principle of Superposition

52

1. Derivation of Equation (38)

Equation (37) was derived from Coulomb's Law and the principle of

superposition:

( ) ∫ ′−

′−′′= 33

0

)(4

1xxxxxxdxE rr

rrrrr

ρπε

. (37)

Taking the divergence of this integral equation with respect to the unprimed

coordinates (coordinates of the observation point where the electric field is being

defined), recognize that the divergence operator does not act on the primed

coordinates:

( ) ∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′−

′−⋅∇′′=⋅∇ 3

3

0

)(4

1xxxxxxdxE rr

rrrrr

ρπε

, (A1)

where ∇ is the gradient with respect to the unprimed coordinates. The divergence in

parenthesis must be determined from an auxiliary integral.

By the divergence theorem:

∫∫=′−≤′− ′−

′−⋅=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′−

′−⋅∇

RxxSRxxV xxxxsd

xxxxxd

rrrrrr

rrr

rr

rr

:3

:3

3 . (A2)

Making the substitution rxx rrr=′− , the surface integral may easily be evaluated:

πφθθφθθπ

θ

π

φ

π

θ

π

φ

4ˆˆsin

ˆˆsin0

2

03

2

0

2

03

2 =⋅=⋅ ∫ ∫∫ ∫= == = R

RnnddRRRnnddR , (A3)

where n represents the unit normal vector to the spherical surface of integration. This

result is valid for all values of R > 0. This condition is met for all observation points

xx ′≠rr . Examining the value of the divergence at xr directly using the same substitution:

011ˆ2

2223 =⎟

⎠⎞

⎜⎝⎛

∂∂

=⋅∇=⋅∇r

rrrr

rrrr . (A4)

53

Under the same condition that xx ′≠rr . Therefore, this divergence must satisfy two

properties, namely:

03 =′−

′−⋅∇

xxxxrr

rr

, if xx ′≠rr , (A5)

and

π433 =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′−

′−⋅∇∫

V xxxxxd rr

rr

, if V contains x′r . (A6)

This gives, by definition, the 3-dimensional Dirac-Delta function:

( )xxxxxx rrrr

rr′−=

′−

′−⋅∇ δ

π 341 . (A7)

Therefore:

( ) ( )( ) ( )xxxxxdxE rrrrrrρ

εδρ

ε 0

3

0

1)(1=′−′′=⋅∇ ∫ , (38)

which is Gauss' Law for electrostatics.

2. Derivation of Equation (39)

Taking the curl of equation (37):

( ) ( )∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′−

′−×∇′′=×∇ 3

3

041

xxxxxxdxE rr

rrrrr

ρπε

. (A8)

Examining the curl in parenthesis directly by arbitrarily considering the x component:

333 ryy

zrzz

yxxxx

x

′−∂∂

−′−

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

′−

′−×∇ rr

rr

( ) ( ) 33

11rz

yyry

zz∂∂′−−

∂∂′−=

54

( ) ( )zr

ryy

yr

rzz

∂∂−′−−

∂∂−′−= 44

33

( ) ( ) 03344 =

′−′−−′−′−=

ryy

rzz

rzz

ryy . (A9)

The other two components of the curl are similarly found to vanish. This is not a

conditional result; therefore, the curl in parenthesis is identically 0. This gives equation

(39),

( ) 0=×∇ xE rr, (39)

which says that the electrostatic field is a conservative vector field.

55

APPENDIX B

Demonstration of Charge Density as the 0-component of a 4-vector

56

Charge is assumed to be a conserved scalar. A differential element of charge,

dq , is related to a static charge density, ρ , by the relationship:

xddq 3ρ= , (B1)

where xd 3 is a differential element of 3-dimensional spatial volume. Since the charge

is a scalar, the R.H.S. of this equation must be a scalar, but the differential volume

element of 3-dimensional space is not a scalar:

3213 dxdxdxxd ≡ . (B2)

There is an element of 4-dimensional space-time volume that is a scalar:43

xddxdxdxdxdxxd 3032104 =≡ . (B3)

To show that equation (B3) represents a scalar (it is not directly obvious from the

discussions in Chapter 2), consider the element of volume under a coordinate

transformation.

32104 xdxdxdxdxd ′′′′=′ . (B4)

From multivariable calculus, this relates to the unprimed element of volume as

xdJxd 44 =′ , (B5)

where is the determinant of the coordinate transformation matrix (a.k.a. the Jacobian).44

For the purposes of Chapter 2, the transformation matrix is the Lorentz transformation

matrix. Without loss of generality, the 1x -axis may be aligned with the Lorentz boost.

Then, using equation (18), the Jacobian of such a boost is

111

100001000000

2

2222 =

−−

=−=−

−

=ββγβγ

γβγβγγ

J . (B6)

Inserting this result into equation (B5) gives

57

xdxd 44 =′ . (B7)

Upon comparison to equation (19), equation (B7) shows that the element of

space-time volume is a scalar. Rearranging equation (B3) gives

03

4

dxxdxd= , (B8)

so this scalar density is the zero component of a 4-vector. This can be generalized to

( ) ( )µµ

xAxd

xdf 03 = , (B9)

where ( )µxdf is an arbitrary scalar field and

( ) ( ) ( ) 04

00 dxxd

xdfdxxgxAµ

µµ == (B10)

is the 0-component of an arbitrary 4-vector field. This shows that the division of an

arbitrary scalar field by a differential element of 3-dimensional spatial volume, which is a

scalar density, results in the 0-component of a 4-vector field. Applying this to electric

charge:

03 jcxd

dqc== ρ . (B11)

This shows that the electric charge density is the 0-component of a 4-vector field.

This 4-vector field is conventionally given the symbol µj and called the charge-current

density 4-vector. The other three components of this 4-vector are the components of

the electric current density 3-vector.

58

APPENDIX C

Antisymmetry of the Faraday Tensor

59

1. Primary Approach

By definition, the components, µνS , of a second rank tensor are symmetric if

νµµν SS = (C1)

and the components, µνA , of a second rank tensor are antisymmetric if

νµµν AA −= (C2)

This definition also holds for covariant and mixed components.

According to equation (48):

0=∂∂ µνµν F . (48)

In special relativity, the twice-repeated partial derivatives are symmetric covariant

components of a second rank tensor. They are symmetric because the result of the

differentiation is the same under an exchange of subscripts. That is:

[ ] [ ]⋅∂∂=⋅∂∂ νµµν . (C3)

As a consequence of the construction of tensors, the contraction on both indices of the

symmetric covariant components of a second rank tensor with the antisymmetric

contravariant components of an second rank tensor vanishes. To show this, let µνS be

the symmetric covariant components of a second rank tensor, and let µνA be the

antisymmetric contravariant components of a second rank tensor. Contracting on both

indices gives

( )( ) νµνµ

νµνµ

µνµν ASASAS −=−= , (C4)

where equations (C1) and (C2) have been used. Recognizing that both of the indices

that appear in equation (C4) are dummy indices, they may be substituted with any

60

arbitrary Greek letter without changing the validity of the equation in any way, and

without using any identity or definition. Choosing the mutual substitution νµ ↔ gives

µνµν

µνµν ASAS −= . (C5)

The only way that equation (C5) can be true is that the contraction definitively vanishes.

A second rank tensor has 16 independent components, in general, whereas

there are only 6 independent antisymmetric components. Since the electric and

magnetic field vectors contribute six independent components, the Faraday tensor is

intuitively antisymmetric, because any extra independent components would bring

superfluous information.9

2. An Alternative Approach

Any second rank tensor may be written as the sum of a symmetric second rank

tensor and an antisymmetric second rank tensor, much like a matrix in linear algebra

can be broken into a symmetric and antisymmetric matrix. So, let the Faraday tensor

be:

µνµνµν SAF += , (C6)

where µνA is the antisymmetric part and µνS is the symmetric part. Inserting this into

equation (48):

[ ] 0=∂∂+∂∂=+∂∂=∂∂ µνµν

µνµν

µνµνµν

µνµν SASAF . (C7)

The term containing the antisymmetric part µνA is identically zero. Therefore,

the symmetric part µνS must satisfy:

0=∂∂ µνµν S . (C8)

The simplest solution to this second order partial differential tensor equation is:9 23

61

0=µνS . (C9)

3. Another Consideration

If the Lorentz force is assumed to be a "pure force," that is, a force that does not

affect the rest mass m of the particle on which it acts, then45

ττµµµµ

ddp

pmd

dpu 1

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ττµµ

µ

µ

ddp

ppddp

m21

( )µµ

τpp

dd

m21

=

( )( ) 021 2 == mc

dd

m τ,

where µu is the proper velocity, or 4-velocity of the particle, and the invariance of

( )2mcpp =µµ has been used. Given that the Lorentz force is of the form:

νµν

µ

τukF

ddp

= ,

it follows that:

0=µνµν uuF .

It should be obvious that the tensor product of the 4-velocity with itself is a

symmetric second rank tensor (because the order of the tensor product does not

change the value of the components). Therefore, from the result in the first section of

this appendix it can clearly be seen that the Faraday tensor must be antisymmetric to

satisfy the above relationship (that is, for the Lorentz force to be a pure force).

62

APPENDIX D

Explicit Calculation of Equation (50)

63

The 1=ν component of equation (47) is

1

0

1 1 jc

Fε

µµ =∂ . (D1)

Writing out all of the terms in equation (D1) explicitly gives

1

03

31

2

21

1

11

0

01 1 jcx

FxF

xF

xF

ε=

∂∂

+∂∂

+∂∂

+∂∂ . (D2)

Using the definitions from Chapter 2 for µx , µj , and µνF , equation (D2) may be written

as

( )( )

( )x

x jcz

Fy

Fxct

E

0

3121 10ε

=∂∂

+∂∂

+∂∂

+∂−∂ . (D3)

Rearranging equation (D3) gives

xx j

zFc

yFc

tE

0

3121 1ε

=∂∂

+∂∂

+∂∂

− . (D4)

A further rearrangement of equation (D4) gives

( ) ( )t

Ej

zcFc

ycFc x

x ∂∂

+=∂

∂+

∂∂

0

312

212 1//

ε, (D5)

and, finally,

t

Ej

cF

zcF

yc x

x ∂∂

+=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

0

31212 1

ε, (50)

which is equation (50).

64

APPENDIX E

Demonstration of Equation (85) and Derivation of Poisson's Equation (86) from

Equation (84)

65

1. Demonstration of Equation (85)

Equation (84) gives a 3-dimensional vector field ( )xg rr according to a mass

density distribution ( )x′rρ . This is derived from the superposition of a central field, and

is therefore a conservative field. Since it is conservative, it has a potential function

( )xrΦ that is related by

( ) ( )xxg rrrΦ−∇= , (E1)

where the gradient operates on the unprimed coordinates.

Since the gradient does not operate on the primed coordinates, it can be taken

inside the integration in equation (85) to operate on only the unprimed coordinates:

( ) ( )∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡′−

∇′′=Φ∇−xx

xxdGx rrrr 13 ρ (E2)

( )( ) ( ) ( )∫ ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

′−+′−+′−∇′′=

222

3 1

zzyyxxxxdG rρ

( ) ( )( ) ( ) ( )( )∫ ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡+

′−+′−+′−

′−−′′= L

r

23222

3 ˆzzyyxx

xxxxxdG ρ ,

where addition of the y and z vector terms are implied,

( ) ( ) ( ) ( )∫ ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

′−

′−+′−+′−′′−= 33 ˆˆˆ

xxzzzyyyxxxxxdG rr

rρ

( ) ( ) ( )∫ ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

′−

′+′+′−++′′−= 33 ˆˆˆˆˆˆ

xxzzyyxxzzyyxxxxdG rr

rρ

( )∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡

′−

′−′′−= 33

xxxxxxdG rr

rrrρ , (84)

66

which is the R.H.S. of equation (84).

2. Derivation of Equation (86)

To get equation (86), the demonstration is analogous to that in Appendix A for

deriving equation (38) from equation (37). The differences are the following two explicit

direct replacements:

( )rE rrΦ−∇→ , (E3)

Gπε

41

0

−→ , (E4)

and to replace the charge density from Appendix A with a mass density in this case.

The negative sign before the constant of proportionality is due to the fact that

gravitational force is always attractive between mass (a mass at the origin exerts a force

on another mass in the r− direction).

67

APPENDIX F

Demonstration of the Partial Derivative of a 4-vector

68

Consider the operation of the partial derivative on an arbitrary 4-vector:

( )[ ]xAx

A µνν

µ

∂∂

=, . (F1)

Now consider an arbitrary coordinate transformation to primed coordinates:

( )[ ] ( ) ⎥⎦

⎤⎢⎣

⎡∂′∂

∂∂

′∂∂

=′′′∂∂ xA

xx

xxxxA

xµ

µ

α

νβ

να

β . (F2)

If the space-time is flat, and the transformation is from one Cartesian coordinate

system to another (a Lorentz transformation), then the partial derivatives of the

coordinates are constant, which allows them to be pulled out of the partial derivative of

the 4-vector:

( ) ( )[ ] νµν

βµαµ

µα

νν

βµ

µ

α

νβ

ν

,AaaxAax

axAxx

xxx

=∂∂

=⎥⎦

⎤⎢⎣

⎡∂′∂

∂∂

′∂∂ . (F3)

Therefore:

νµν

βµα

βα

,, AaaA =′ . (F4)

This has the form of equation (31), and therefore the operation of the partial derivative

on a four vector is a tensor product that results in a second rank tensor if there is no

contraction, or a scalar if there is contraction. This idea was used more than once in

Chapter 2.

If, however, the space-time is not flat, then the partial derivatives of the

coordinates are not constant. According to the product rule for differentiation:

( ) ( )[ ] ( )xAxx

xxxxA

xxx

xxxA

xx

xxx µ

µν

α

β

νµ

νµ

α

β

νµ

µ

α

νβ

ν

∂∂′∂

′∂∂

+∂∂

∂′∂

′∂∂

=⎥⎦

⎤⎢⎣

⎡∂′∂

∂∂

′∂∂ 2

µµν

α

β

ν

νµ

µ

α

β

ν

Axx

xxxA

xx

xx

∂∂′∂

′∂∂

+∂′∂

′∂∂

=2

, . (F5)

69

It is this second term that appears on the R.H.S. that prevents the ordinary partial

derivative operation from resulting in a tensor in curved spacetime.

This demonstration is shown by Ohanian and most other references that use

tensor analysis to discuss space-time curvature.33

To show how the Christoffel symbol subtracts this problem from the

transformation, consider the transformation of the covariant derivative using equation

(88) and (89).

σασβ

αββ

α AAA ′Γ′+′∂′=′ ; . (F6)

The form of the first term on the R.H.S. of equation (F6) has already been shown in

equation (F5). To find the form of the second term, note the transformation property of

the Christoffel symbol37

βσ

µ

µ

αµλνβ

ν

σ

λ

µ

αα

σβ xxx

xx

xx

xx

xx

′∂′∂∂

∂′∂

+Γ′∂

∂′∂

∂∂′∂

=Γ′2

, (F7)

which is a consequence of applying a coordinate transformation explicitly to the R.H.S.

of equation (89) and then simplifying. Equation (F6) becomes

µµν

α

β

ν

νµ

µ

α

β

ν

βα A

xxx

xxA

xx

xxA

∂∂′∂

′∂∂

+∂′∂

′∂∂

=′2

,;

κκ

σ

βσ

µ

µ

αµλνβ

ν

σ

λ

µ

α

Axx

xxx

xx

xx

xx

xx

∂′∂

⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂

∂∂′∂

+Γ′∂

∂′∂

∂∂′∂

+2

. (F8)

Rearranging

κκ

σ

σ

λµλνβ

ν

µ

α

νµ

µ

α

β

ν

βα A

xx

xx

xx

xxA

xx

xxA

∂′∂

′∂∂

Γ′∂

∂∂′∂

+∂′∂

′∂∂

=′ ,;

κβσ

µ

µ

α

κ

σµ

µν

α

β

ν

Axx

xxx

xxA

xxx

xx

′∂′∂∂

∂′∂

∂′∂

+∂∂′∂

′∂∂

+22

. (F9)

70

Recognizing the product of the transformation matrix with its own inverse as the

Kronecker delta tensor in the second term, equation (F9) becomes

λµλνβ

ν

µ

α

νµ

µ

α

β

ν

βα A

xx

xxA

xx

xxA Γ

′∂∂

∂′∂

+∂′∂

′∂∂

=′ ,;

κβσ

µ

µ

α

κ

σµ

µν

α

β

ν

Axx

xxx

xxA

xxx

xx

′∂′∂∂

∂′∂

∂′∂

+∂∂′∂

′∂∂

+22

. (F10)

Consider the partial derivative of the Kronecker delta tensor, which vanishes in

any coordinate system.

0=′∂∂

βα

σ δx. (F11)

The Kronecker delta tensor can be replaced by the product of the coordinate

transformation matrix with its own inverse.

βσ

µ

µ

α

β

µ

µσ

α

β

µ

µ

α

σ xxx

xx

xx

xxx

xx

xx

x ′∂′∂∂

∂′∂

+′∂

∂∂′∂′∂

==⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂∂′∂

′∂∂ 22

0 . (F12)

Using the chain rule

µλ

α

σ

λ

β

µ

βσ

µ

µ

α

xxx

xx

xx

xxx

xx

∂∂′∂

′∂∂

′∂∂

−=′∂′∂

∂∂′∂ 22

. (F13)

Substituting equation (F13) into equation (F9) gives

λµλνβ

ν

µ

α

νµ

µ

α

β

ν

βα A

xx

xxA

xx

xxA Γ

′∂∂

∂′∂

+∂′∂

′∂∂

=′ ,;

κµλ

α

σ

λ

β

µ

κ

σµ

µν

α

β

ν

Axx

xxx

xx

xxA

xxx

xx

∂∂′∂

′∂∂

′∂∂

∂′∂

−∂∂′∂

′∂∂

+22

λµλνβ

ν

µ

α

νµ

µ

α

β

ν

Axx

xxA

xx

xx

Γ′∂

∂∂′∂

+∂′∂

′∂∂

= ,

κκ

λµλ

α

β

µµ

µν

α

β

ν

δ Axx

xxxA

xxx

xx

∂∂′∂

′∂∂

−∂∂′∂

′∂∂

+22

71

λµλ

α

β

µµ

µν

α

β

νλµ

λνβ

ν

µ

α

νµ

µ

α

β

ν

Axx

xxxA

xxx

xxA

xx

xxA

xx

xx

∂∂′∂

′∂∂

−∂∂′∂

′∂∂

+Γ′∂

∂∂′∂

+∂′∂

′∂∂

=22

, . (F14)

Utilizing the freedom to reassign contracted indices and the fact the repeated partial

derivatives commute gives

( ) µµν

α

β

ν

µν

α

β

νλµ

λννµ

β

ν

µ

α

βα A

xxx

xx

xxx

xxAA

xx

xxA ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂′∂

′∂∂

−∂∂′∂

′∂∂

+Γ+′∂

∂∂′∂

=′22

,;

νµ

µ

α

β

ν

;Axx

xx

∂′∂

′∂∂

= . (F15)

Therefore, according to the rules set forth in Section 2.3, the form of equation (F15)

shows that the covariant derivative of a 4-vector is a tensor of second rank. By

induction (though not formally), the covariant derivative operator itself is a covariant 4-

vector, or a 4-covector.

µνν

µ ADA =; . (F16)

Combining equations (F15) and (F16) gives

νβ

ν

β DxxD′∂

∂=′ . (F15)

72

APPENDIX G

Derivation of Equation (99) from Equation (94)

73

Equation (94) says that a geodesic maximizes the proper time:

max=∫geodesic

dτ . (G1)

From Chapter 1 Sec 3 the definition of proper time leads to:

( )∫∫∫ ++== lkkl

kk dxdxgdtcdxgdtcg

cdxdxg

cd 0

200 211 νµ

µντ

∫∫ ++=++= lkkl

kk

lk

kl

k

k gggdtdtc

dxdtc

dxgdtc

dxggdt βββ000000 22 , (G2)

where dtc

dxkk =β , similar to the definition in equation (11). If this integration is carried

out along a geodesic, then

max2 000 =++∫geodesic

lkkl

kk dtggg βββ . (G3)

Multiplying2 by 2cm− gives

min2 0002 =++− ∫

geodesic

lkkl

kk dtgggcm βββ . (G4)

Notice that the integral is now minimized due to the negative sign. Assuming that the

metric tensor deviates by a small amount from the Minkowski Metric tensor, and that the

velocities are slow compared to the speed of light, the square root can be

approximated.2

( )∫∫ −−−−−=geodesic

lkkl

kk

geodesic

dtgggcmd βββτ 0002 211

( )( )∫ −−−−−≈geodesic

lkkl

kk dtgggcm βββ0002

12 211

( )( )∫ −−−−≈geodesic

kk dtgcm ββ002

12 11 . (G5)

74

where the l superscript in the last term has been lowered by the metric tensor and the

raised and lowered Latin index k indicates a summation from 1 to 3. The radical has

been expanded assuming that the parenthetical is 1<< , and, aside from the 00g

component, the metric tensor is approximated as the Minkowski metric tensor.2

Simplifying, and making use of equation (11), equation (G5) becomes

( )∫∫ −+−−=geodesicgeodesic

dtcmgcmcmcmd 2221

002

212

212 βτ

( )∫ +−+−=geodesic

dtvmgcmcmcm 221

002

212

212

( )( ) min221

002

212

21 =+−= ∫ dtcgcmmv

geodesic

, (99)

upon rearranging, which is equation (99).

75

APPENDIX H

The Stress-energy Tensor from the "Cloud of Dust" Model

76

Imagine an extremely dense (in the continuum limit) cloud of identically massive

dust particles (point masses) of rest mass m , but that they do not interact with each

other. This is the popular "cloud of dust" model used to demonstrate the stress-energy

tensor.41 In some small element of spatial volume V∆ , there is some number of

particles n , each with a rest mass m . For simplicity, consider all of the particles to be

co-moving. Then, the momentum in this element of volume is

µµ umnp = . (H1)

The energy of this element of the dust cloud is given as

cumncpE 00 ==∆ . (H2)

Therefore, the energy density in this volume is

cuVmn

VE 0

∆=

∆∆ . (H3)

In the limit that 0→∆V , equation (H3) gives

( ) 00000003

~1 Tuucujc

cuxdmnd

≡== ρ , (H4)

where ρ~ is the rest frame mass density of the dust cloud and 00T is defined as the

energy density of the matter field (i.e. the density of 2cm ). The appearance of 0j is by

a similar argument to that found in Appendix B, except that here it indicates the 0-

component of the mass current density 4-vector.

In order to form a tensor, all that must be done is to generalize the indices from

0 to µ and ν . Then, equation (H4) is still identically valid in the rest frame, and

furthermore, validity is maintained under a Lorentz boost.

νµµν ρ uuT ~≡ . (H5)

77

Lowering the second index of equation (H5) for the 00 component gives

ααρ uugT 0

000 ~= . (H6)

In the static weak field limit, 00 αα η→g , and cu →0 . Therefore equation (H6) gives

200000

0 ~~~ cuuuuT ρρηρ αα →=→ . (H7)

For an interpretation of this new tensor, consider three sets of components, 00T ,

0kT , and klT .

00T is the energy density,41 as before. It is the amount of energy per unit volume

by virtue of the presence of rest mass, whether or not the mass is in motion.

kk TT 00 = is the momentum density, or energy flux density.41 This is quite similar

to current density. It is due to a stream of massive point particles flowing in the k

direction (or, at least, the flow has a projection in the k direction).

lkkl TT = is the momentum flux density of k directed momentum transferred in

the l direction.41 This is similar to a transverse wave in the case that lk ≠ , and a

longitudinal wave in the case that lk = .

As a simple example, consider a constant, continuous, and static distribution of

non-interacting mass. The stress energy tensor has components 200 ~ cT ρ= , and

0=µνT if either 0≠µ or 0≠ν . A Lorentz boosted version of this stress energy tensor

in the x direction is essentially the same as a uniform distribution of mass moving as a

whole in the x direction (resulting in a Lorentz factor due to relativistic velocity addition)

with an increased mass density (due to length contraction and resulting in another

Lorentz factor). This gives 2200 ~ cT ργ=′ , vcTT ργ ~20110 =′=′ , and 2211 ~ vT ργ=′ .

Therefore, the 00 component is still the familiar rest energy density with the relativistic

78

correction factor of 2γ , the 10 component is the classical density of the momentum in

the x direction with the relativistic correction factor of c2γ , and the 11 component is the

classical kinetic energy density with the relativistic correction factor 2γ .

79

APPENDIX I

Generalization of 002 g∇ to 0

02 R−

80

According to the notation of Chapter 2, the Laplacian of 00g can be written as

lkklk

k ggg ,,000000

2 η−=∂−∂=∇ , (I1)

where the summations on k and l are from 1 to 3. The use of Latin indices is allowed

by the diagonal nature of the Minkowski metric tensor, since including the 0 index terms

would not contribute anything to the summation.

The quantity in equation (I1) must be related to a tensor in order to be a physical

object. As it stands, the object in equation (I1) has two free indices, both 0. So, it is

reasonable to generalize to the 00 component of a second rank tensor. The most

obvious first attempt is to simply generalize the ordinary partial derivatives summed

from 1 to 3 to covariant derivatives summed from 0 to 3. This, by definition, would make

the object the 00 component of a tensor directly in a single step.

µµ

;;

00,,

00 gg kk −→ . (I2)

Unfortunately, this is not an acceptable candidate, since this object, also by definition,

vanishes when 00g is interpreted as a component of the general metric tensor.

0;00 =µg (I3)

The next attempt (not at all obvious, and not nearly as direct) is to transpose the

indices in order to account for all of the components of the metric tensor. Consider the

explicit summation in equation (I1).

∑∑= =

−=∇3

1

3

1

,,0000

2

k l

lkkl gg η (I4)

Transposing the indices on lkg ,,00 gives

( )∑∑= =

+++−=∇3

1

3

1

0,,0

0,0,,0,0

,,0000

2

k l

klkl

lk

lkkl ggggg η . (I5)

81

The equality remains in equation (I5) in the static limit since all three additional terms

contain a derivative with respect to time and therefore vanish. This accounts for all of

the components of the general metric tensor. The operation of raising an lowering can

be done implicitly by accounting for a change of sign when the operation involves a

Latin index. This gives

( )( ) ( )( ) ( ) ( )( )∑∑= =

↓↓↓↓↓↓↓↓↑↑ −++−+−−−−−=∇

3

1

3

10,,00,0,,0,0,,0000

2 111111k l

klkllklkkl ggggg η

( )∑∑= =

−+−−=3

1

3

10,,00,0,,0,0,,00

k lklkllklk

kl ggggη (I6)

The Latin indices can be generalized to Greek indices also without changing the validity

of equation (I6) in the static limit, since such a generalization will only introduce time

derivatives which vanish in the static limit. The explicit summation is now removed in

favor of notational convenience over functional explicitness.

( )0,,00,0,,0,0,,00002

µνµννµνµµνη ggggg −+−−=∇ . (I7)

Two mutually canceling terms may be added to equation (I7), namely µν ,0,0g− and

µν ,0,0g . Inserting these two terms and rearranging gives

( )νµµνµννµµνµνµνη ,0,00,0,,0,0,,000,,0,0,000

2 ggggggg −+++−−−=∇ . (I8)

Using the symmetry of the metric tensor and the commutability of the ordinary partial

derivatives, equation (I8) may be rewritten.

( )0,,00,0,0,,0,,00,0,0,0,0002

νµµνµνµνµνµνµνη ggggggg −+++−−−=∇ . (I9)

Grouping the first three terms together and the last three terms together and

recognizing the Minkowski metric tensor as a constant gives

( )[ ] ( )[ ]νµµνµνµν

νννµν

µ ηη ,00,,00,000,00,0002 ggggggg −+∂−−+∂=∇ . (I10)

82

In local geodesic coordinates µνµν η=g at the pole of these coordinates. In such

coordinates, equation (I10) may be written as

( )[ ] ( )[ ]νµµνµνµν

νννµν

µ ,00,,00,000,00,0002 ggggggggg −+∂−−+∂=∇ . (I11)

The quantities in square brackets are recognized as constant scalar multiples of the

Christoffel symbol according to equation (89).

[ ] [ ] ( )µµ

µµ

µµ

µµ 0,0,00000000

2 222 Γ+Γ−−=Γ∂−Γ∂=∇ g . (I12)

In geodesic coordinates, the quantity in parenthesis of equation (I12) is recognized as

exactly the 00R component of the Ricci tensor according to equations (90) and (91).

00002 2 Rg −=∇ . (I13)

The first subscript of this component of the Ricci tensor may be raised with the metric

tensor. In geodesic coordinates 00 αα η→g , therefore, raising and lowering a 0 index

does not change the value of the tensor component. This gives (numerically)

00

002 2 Rg −=∇ . (I14)

The justification for using geodesic coordinates is in the weak field limit.

Geodesic coordinates are coordinates for which the metric tensor is locally flat, and

therefore can be replaced by the Minkowski metric tensor. This is representative of the

metric that would be observed, for instance, inside an elevator that is freely falling in the

Earth's gravitational field. Essentially, a freely falling Cartesian coordinate system is a

geodesic coordinate system in a weak gravitational field for some small amount of time.

At this point, it should be emphasized that the equality in equation (I10) is exact

in the static limit, and that, by choosing a geodesic coordinate system, the

generalization from (I10) to (I11) is locally exact. A transformation from the R.H.S. of

83

equation (I11) to a general coordinate system will generate two extra terms that are

products of Christoffel symbols. These two terms match those found in the definition of

the Riemann tensor so that the R.H.S. of equation (I11) is found to transform as a

tensor in general coordinates.

84

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42 A. Einstein, The Meaning of Relativity, 3rd ed. (Princeton Univ. Press, Princeton, NJ,

1950), pp. 50-2.

87

43 A. Einstein, The Meaning of Relativity, 3rd ed. (Princeton Univ. Press, Princeton, NJ,

1950), p. 68.

44 J. Stewart, Calculus: Early Vectors, preliminary ed. (Brooks/Cole Publ. Co., Pacific

Grove, CA, 1997), Vol. III, pp. 71-7.

45 W. Rindler, Relativity: Special, General and Cosmological (Oxford University Press,

Inc., New York, 2001), pp. 123-5.